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Linear Programming
Published in Anindya Ghosh, Prithwiraj Mal, Abhijit Majumdar, Advanced Optimization and Decision-Making Techniques in Textile Manufacturing, 2019
Anindya Ghosh, Prithwiraj Mal, Abhijit Majumdar
LP methods are some of the most popular, versatile, and commonly used approaches that provide optimal solutions to the problems with linear objective and constraint functions. This chapter presents a detailed application of graphical and simplex methods of solving linear programs with suitable examples related to textile manufacturing, along with coding in MATLAB® language. In the graphical method, initially a feasible region is identified in a two-dimensional plane, after satisfying all of the constraints simultaneously. Then, by a trial-and-error method, a point is located in the feasible region whose coordinate gives the optimum value. Though the graphical method is easy to understand, it can only be used when the number of variables in a linear problem is limited to two. Simplex methods are found to be very efficient in solving LP problems of any magnitude. The simplex method begins with a basic feasible solution and at each iteration it projects an improved solution over the earlier step. A solution is considered optimum when there is no further improvement. Simplex methods require formulation of the LP in a standard form in which all the inequality constraints are converted into equality constraints by using slack, surplus, or artificial variables. Then key operations at each iteration are performed to obtain the optimal solution. Published works on application of LP in textile manufacturing are also illustrated in this chapter.
Modeling Processes and Operations with Linear Programming
Published in Robert M. Peart, R. Bruce Curry, Agricultural Systems Modeling and Simulation, 2018
To solve using the simplex procedure, the constraint inequalities are corrected to equalities via the addition of slack variables. These slack variables represent unused land (x4), unused labor (x5), and unused capital (x6). x1+x2+x3+x4=1003x1+4x2+5x3+x5=400300x1+260x2+230x3+x6=25,000
Linear Programming
Published in Michael W. Carter, Camille C. Price, Ghaith Rabadi, Operations Research, 2018
Michael W. Carter, Camille C. Price, Ghaith Rabadi
The Simplex method is a general solution method for solving linear programming problems. It was developed in 1947 by George B. Dantzig and, with some modifications for efficiency, has become the standard method for solving very large linear programming problems on computers. Most real problems are so large that a manual solution via the Simplex method is impractical, and these problems must be solved with Simplex programs implemented on a computer. Small problems, however, are quite useful in demonstrating how the Simplex method operates; therefore, we will use such problems to illustrate the various features of the method.
Optimizing the Implementation of Small Modular Reactors into Ontario’s Future Energy Mix
Published in Nuclear Technology, 2023
C. Colterjohn, S. Nagasaki, Y. Fujii
A subset of mathematical programming, linear programming is one of the simplest methods of optimization. Given a set of decision variables and subject to constraints defined by linear equalities and/or inequalities, a linear objective function may be optimized to find its maximum or minimum values. A feasibility region is first created by the function, whose boundaries are defined by the linear constraints, and each point along which represents a possible solution corresponding to varying values for the decision variables. In most practical situations, the number of decision variables can grow very quickly depending on the system that is being modeled. The Simplex method is an iterative process for solving linear programming problems, where the values of the decision variables are sequentially tested until an optimal solution is found. This method involves writing the objective function in the form
Improving a primal–dual simplex-type algorithm using interior point methods
Published in Optimization, 2018
T. Glavelis, N. Ploskas, N. Samaras
Linear Programming (LP) is a significant research area in the field of operations research. Linear programs can be found in almost every type of scientific and engineering applications. The first approach for solving linear programming problems (LPs) came from George B. Dantzig who set the fundamental principles. Dantzig proposed the simplex algorithm [1] that starts from a basic feasible solution and moves from one basic feasible solution to an adjacent one until an optimum solution is found. The vast literature of operations research contains an extensive family of simplex-type algorithms. Although the simplex algorithm is widely-used until today, there exist many real-life applications that its performance degrades due to a phenomenon called degeneracy. Simplex-type algorithms may stall at a degenerate vertex for many iterations before moving to another vertex. Degeneracy and redundancy are very common in real-life applications. Consequently, many different approaches were presented in order to overcome degeneracy [2]. Over the last decades, researchers focused on: (i) the reduction of the number of iterations that the simplex algorithm performs [3–5] (for a review, see [6]), (ii) the reduction of the computational work involved in each iteration [7–10] (for a review, see [11]), and (iii) new variants of the simplex algorithm [12–15].
International portfolio optimization based on uncertainty theory
Published in Optimization, 2021
It is obvious that the model (15) is a linear programming model. For linear programming problem, simplex method is a proved method to solve its solution. So we give the analytical solution of the model (15) according to the simplex method. For model (15), we assume that consists of and . Since the rank of the constraint coefficient matrix is 2 (), is a 2-vector made up of basic variables and is a zero vector consisting of n−2-dimensional nonbasic variables.